Commutative algebra and algebraic geometry may be thought of as studying solutions of many equations in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but it is often best encoded into a family of functions defined on this set. The abstract version of such families of functions are called commutative rings. Homological algebra brings to the study of rings methods of algebraic topology, developed to study geometric structures. Methods for decomposing complicated structures into primitive building blocks are developed as part of a different branch of algebra, known as representation theory.
Avramov will investigate problems arising at a crossroads of commutative algebra, homological algebra, and representation theory. The combination of different points of view allows for a fruitful transfer of techniques and intuition between fields.
Properties of commutative noetherian rings will be studied through numerical, algebraic, and geometric invariants generated by homological constructions. These include free resolutions, Hochschild cohomology, stable cohomology, cohomological support varieties, derived categories of modules and of differential modules. Special attention will be paid to developing finitistic recognition criteria for geometrically important properties of families of commutative rings. Connections through homological algebra with representation theory of finite dimensional algebra will be studied.
The main results obtained under the award include, but are not limited to, the following: Properties of cohomology of modules over local complete intersection rings were deduced from properties of modules over exterior algebras (which are much easier to study). Poincare series of all modules over generic short Gorenstein rings were shown to admit an explicit common denominator. Grothendieck duality theory was applied to study reflexivity and rigidity of complexes of modules and complexes of sheaves. A new construction of commutative rings, called their connected sum, was introduced and applied to produce unusual Gorenstein rings. A conjecture of Vasconcelos, characterizing flatness over smooth bases, was proved. Far-reaching generalization were obtained of a classical criterion for regularity in positive characteristic, due to Kunz. Sharp bounds were obtained for the area where the minimal free resolution of a Koszul algebra over a polynomial ring is located. Bass numbers of rings of embedding codepth at most 3 were shown to grow strictly exponentially. Gorenstein modules over rings admitting a contracting endomorphism were characterized. The structure of minimal free resolutions of ideals generated by 4 bihomogeneous variables was described. Among the broader impact of work done under this award is extensive training of graduate students and post-docs. Funds from the award were used to in invite graduate students and post-docs from other institutions.
